What is Audit Sampling? Applying a procedure to less than 100% of a population To estimate some characteristic of the population Qualitative Quantitative

Risk Sampling risk risk that the auditors’ conclusions based on a sample may be different from the conclusion they would reach if they examined every item in the population Nonsampling risk risk pertaining to nonsampling errors Can be reduced to low levels through effective planning and supervisions of audit engagements

Nonstatistical Sampling The auditor estimates sampling risk by using professional judgment rather than statistical techniques Provides no means of quantifying sampling risk Sample may be larger than necessary or auditors may unknowingly accept a higher than acceptable degree of sampling risk

Advantages of Statistical Sampling Allows auditors to measure and control sampling risk which helps: Design efficient samples Measure sufficiency of evidence Objectively evaluate sample results 2

Selection of Random Sample Random sample results in a statistically unbiased sample that may not be a representative sample Random sample techniques Random number tables Random number generators Systematic selection 3

Random Number Table

Other Methods of Sample Selection Haphazard selection Select items on an arbitrary basis, but without any conscious bias Block selection Block sample consists of all items in a selected time period, numerical sequence or alphabetical sequence Stratification Technique of dividing population into relatively homogeneous subgroups

An Illustration of Stratification

Types of Statistical Sampling Plans Attributes sampling Discovery sampling Classical variables sampling Mean-per-unit estimation Ratio estimation Difference estimation Probability-proportional-to-size sampling 4

Dual Purpose Test Tested used both as a test of control and substantiating the dollar amount of an account balance Ex. Test to evaluate the effectiveness of a control over recording sales transactions and to estimate the total overstatement or understatement of the sales account

Allowance for Sampling Risk Amount used to create a range, set by + or – limits from the sample results, within which the true value of the population characteristic being measured is likely to lie Precision Wider the interval, more confident but less precise conclusion Can be used to construct a dollar interval

Sample Size Significant effect on allowance for sampling risk and sampling risk Sample size increase -> sampling risk and allowance for sampling risk decrease Sample size affected by characteristics of population Generally as Population increases -> sample size increase

Requirements of Audit Sampling Plans When planning the sample consider: The relationship of the sample to the relevant audit objective Materiality or the maximum tolerable misstatement or deviation rate Allowable sampling risk Characteristics of the population Select sample items in such a manner that they can be expected to be representative of the population Sample results should be projected to the population Items that cannot be audited should be treated as misstatements or deviations in evaluating the sample results Nature and cause of misstatements or deviations should be evaluated 5 5

Sampling Risks--Tests of Controls Actual Extent of Operating Effectiveness of the Control Procedure is Adequate Inadequate The Test of Controls Sample Indicates: Extent of Operating Effectiveness is Adequate Effectiveness Inadequate Correct Decision Incorrect Decision (Risk of Assessing Control Risk Too Low) Incorrect Decision (Risk of Assessing Control Risk Too High) Correct Decision 6 6

Audit Sampling Steps for Tests of Controls Determine the objective of the test Define the attributes and deviation conditions Define the population to be sampled Specify: The risk of assessing control risk too low The tolerable deviation rate Estimate the population deviation rate Determine the sample size Select the sample Test the sample items Evaluate the sample results Document the sampling procedure 7 7

Attributes Sampling: Relationship Between the Planned Assessed Level of Control Risk and the Tolerable Deviation Rate

Illustration of Attributes Sampling--Determining Sample Size Risk of Assessing Control Risk Too Low—5 percent Tolerable Deviation Rate—9 percent Expected Population Deviation Rate—2 percent

Figure 9.4: Statistical Sample Sizes for Tests of Controls at 5 Percent Risk of Assessing Control Risk Too Low

Sample Size Sample size using Figure 9-4 (next slide) =68 (2) This means the auditor should select a sample of 68 items. We will discuss the (2) in a few slides.

Attributes Sampling Evaluation of Results 2 possible approaches: Use the bracketed number from Table 9.4. If you find that number or less deviations, conclude that you have accomplished your audit objective. Use Table 9.5 for a more precise conclusion.

Example A--No Deviations Identified (Evaluating Attributes Sampling Results) Approach 1—You have met your audit objective (because the bracketed number was (2), you meet objective when you identify 0, 1 or 2 deviations). What can you say? “I believe that the deviation rate in the population is less than 9 percent.” You will be wrong 5 percent of the time when the deviation is exactly 9 percent. If the deviation rate is in excess of 9 percent you will be wrong even less than 5 percent of the time. The planned assessed level of control risk is achieved. Approach 2 You have tested 68 items, a number not on Table 9-5 (next slide To be conservative go to next lowest number on table (65) and use it for your conclusions (we could, but won't interpolate for a more precise answer). You have met your audit objective. Table 9-5 gives us an answer of 4.6 percent. What can you say? "I believe that the deviation rate in the population is less than 4.6 percent.” You will be wrong 5 percent of the time when the deviation rate is exactly 4.6 percent. If the deviation rate is in excess of 4.6 percent you will be wrong even less than 5 percent of the time. The planned assessed level of control risk is achieved.

Figure 9.5 Statistical Sampling Results Evaluation Table for Tests of Controls: Achieved Upper Deviation Rate at 5 Percent Risk of Assessing Control Risk Too Low

Example B--3 Deviations Identified (Evaluating Attributes Sampling Results) Approach 1—You have not met your audit objective. What can you say? “The achieved upper deviation rate is higher than 9 percent.” The planned assessed level of control risk is not achieved. You need to consider increasing the assessed level of control risk above the planned assessed level. Accordingly, you may not “rely” on internal control to the extent planned. Thus, the auditor will need to increase the scope of substantive procedures (the nature, timing, and/or extent). Approach 2—You have not met your audit objective. Table 9-5 provides us an answer of 11.5 percent “I believe that the deviation rate in the population is less than 11.5 percent.” You will be wrong 5 percent of the time when the deviation rate is exactly 11.5 percent. But this is not good enough as you wanted 9 percent rather than 11.5 percent. The planned assessed level of control risk is not achieved. You need to consider increasing the assessed level of control risk above the planned assessed level. As per Approach 1, an increase in the scope of substantive procedures is appropriate.

Other Statistical Attributes Sampling Approaches Discovery sampling Purpose is to detect at least one deviation, with a predetermined risk of assessing control risk too low if the deviation rate in population is greater than specified tolerable deviation rate Useful in suspected fraud Sequential (Stop-or-Go) Sampling Audit sample taken in several stages

Sampling Risks--Substantive Tests The Population Actually is Not Materially Materially Misstated Misstated The Substantive Procedure Sample Indicates Misstatement in Account Exceeds Tolerable Amount Account Is Less Than Tolerable Amount Correct Decision Incorrect Decision (Risk of Incorrect Rejection) Incorrect Decision (Risk of Incorrect Acceptance) Correct Decision 8 8

Audit Sampling Steps for Substantive Tests Determine the objective of the test Define the population and sampling unit Choose an audit sampling technique Determine the sample size Select the sample Test the sample items Evaluate the sample results Document the sampling procedure 9 9

Population Variability—Why it Matters Item Population A Population B 1 2,100 8,000 2 2,100 25 3 2,100 2,000 4 2,100 400 5 2,100 75 Mean 2,100 2,100 Standard deviation -0- 3,395 The variability determines how much information each of the items in the population tells you about the other items in the population. 2 11

Factors Affecting Sample Size

Mean Per Unit (MPU) Illustration Population Size = 100,000 accounts Book value = $6,250,000 Other information: Tolerable misstatement = $364,000 Sampling risk Incorrect Acceptance = 5% Incorrect Rejection = 4.6 %

MPU Risk Coefficients Acceptable Level of Risk (%) Incorrect Acceptance Coefficient Incorrect Rejection Coefficient 1.0 2.33 2.58 4.6 1.68 2.00 5.0 1.64 1.96 10.0 1.28 15.0 1.04 1.44 20.0 .84 25.0 .67 1.15 30.0 .52 40.0 .25 50.0 .00

Determining Sample Size--MPU (1 of 2)

Determining Sample Size--MPU (2 of 2) = 225 Accounts

Variables Sampling Illustration--MPU Adjusted allowance for sampling risk = Tolerable _ (Population size * Incorrect acceptance coef. * Sample stan. dev.) misstatement Sample size This formula “adjusts” the allowance for sampling risk to consider the standard deviation of the audited values in the sample. It holds the risk of incorrect acceptance at its planned level. 4 13

Variables Sampling Illustration--MPU Using the text example with a standard deviation of audited values of $16 Adjusted allowance for sampling risk = Tolerable _ (Population size * Incorrect acceptance coef. * Sample stan. dev.) misstatement Sample size = $364,000 _ ($100,000 * 1.64 * $16) 225 = $189,067 We would still “accept” the book balance because the $6,250,000 (book value) falls within this interval Estimate of total + Adjusted allowance audited value for sampling risk $6,100,000 + $189,067 [$5,910,933 to $6,289,067]

Acceptance Interval Figure 9-12

Difference Estimation Use sample to estimate the avg. difference between the audited value and book value of items in population Projected = Sample Net Misstatement * Pop. Items Misstatement Sample items Most appropriate when size of misstatements does not vary significantly in comparison to book value

Ratio Estimation Use a sample to estimate the ratio of misstatement in a sample to its book value and project it to population Projected = Sample Net Misstatement * Pop. Book Value Misstatement Book Value of Sample Preferred when the size of misstatements is nearly proportional to the book values of the items Large accounts have large misstatements

Nonstatistical Variables Sampling Illustration Plan Sample: Population: Size = 363 items Book value = $200,000 Tolerable misstatement = $10,000 Risk assessments: Inherent and control risk = Slightly below maximum Other substantive tests = Moderate 8 15

Nonstatistical Sampling--Determination of Sample Size Sample size = Population book value X Reliability factor Tolerable misstatement = $200,000 X 2.0 = 40 items $10,000 9 16

Nonstatistical Sampling--Evaluation of Sample Results 40 accounts in sample $350 net overstatement $60,000 book value of sample items Projected misstatement: = [Sample net misstatement] X Book value of population [ Book value of sample ] = [ $350 ] X $200,000 [$60,000] = $1,167 Since the projected misstatement is only 11.7 percent ($1,167/$10,000) of tolerable misstatement, it is likely that the auditors would conclude that the account balance is materially correct. 10 17

PPS Sampling Illustration Population book value = $6,250,000 Other Information: Tolerable misstatement = $364,000 Sampling risk--Incorrect acceptance = 5% Expected misstatement = $50,000 Use Figures 9-14 and 9-15 to obtain a “reliability factor” and an “expansion factor”--next slide 6 18

PPS Sampling Reliability and Expansion Factors

PPS Sample Size Computation Recorded amount of population * Reliability factor Tolerable misstatement - (Expected misstatement * Expansion factor) = $6,250,000 * 3.0 = 66 $364,000 - ($50,000 * 1.6) Sampling interval = Book value of the population Sample size = $6,250,000 = $95,000 (approximately) 66 7 19

Figure 9.16 PPS Sample Selection Process

PPS Evaluation of Results Upper Limit on misstatement = Projected misstatement + Basic precision (Rel. factor x interval) + Incremental allowance

Calculation of Upper Limit on Misstatement

Comparison of statistical sampling techniques for substantive procedures

Audit Risk AR = IR x CR x DR where AR=The allowable audit risk that a material misstatement might remain undetected for the account balance and related assertions. IR= Inherent risk, the risk of a material misstatement in an assertion, assuming there were no related controls. CR= Control risk, the risk that a material misstatement that could occur in an assertion will not be prevented or detected on a timely basis by internal control. DR= Detection risk, the risk that the auditors’ procedures will fail to detect a material misstatement if it exists.